JOURNAL OF MATHEMATICAL PHYSICS 54, 081703 (2013)
Discovering real lie subalgebras of e6 using Cartan
decompositions
Aaron Wangberg1,a) and Tevian Dray2,b)
1
Department of Mathematics and Statistics, Winona State University,
Winona, Minnesota 55987, USA
2
Department of Mathematics, Oregon State University, Corvallis, Oregon 97331, USA
(Received 10 March 2013; accepted 31 July 2013; published online 23 August 2013)
The process of complexification is used to classify Lie algebras and identify their
Cartan subalgebras. However, this method does not distinguish between real forms
of a complex Lie algebra, which can differ in signature. In this paper, we show
how Cartan decompositions of a complexified Lie algebra can be combined with
information from the Killing form to identify real forms of a given Lie algebra. We
apply this technique to sl(3, O), a real form of e6 with signature (52, 26), thereby
identifying chains of real subalgebras and their corresponding Cartan subalgebras
within e6 . Motivated by an explicit construction of sl(3, O), we then construct an
Abelian group of order 8 which acts on the real forms of e6 , leading to the identification
of 8 particular copies of the 5 real forms of e6 , which can be distinguished by
C 2013 AIP Publishing LLC.
their relationship to the original copy of sl(3, O). [http://dx.doi.org/10.1063/1.4818503]
I. INTRODUCTION
The group E6 has a long history of applications in physics, beginning with the original discovery
of the Albert algebra and its relationship to exceptional quantum mechanics.1–3 As a candidate gauge
group for a Grand Unified Theory, E6 is the natural next step in the progression SU(5), SO(10), both of
which are known to lead to interesting, albeit ultimately unphysical, models of fundamental particles
(see, e.g., Ref. 4), and is closely related to the Standard Model gauge group SU(3) × SU(2) × U(1).
A description of the group E6( − 26) as S L(3, O) was given in Ref. 5, generalizing the interpretation of S L(2, O) as (the double cover of) SO(9, 1) discussed in Ref. 6. An interpretation combining
spinor and vector representations of the Lorentz group in 10 spacetime dimensions was described in
Ref. 7, and in Ref. 8 we obtained nested chains of subgroups of S L(3, O) that respect this Lorentzian
structure.
The resulting action of E6 appears to permit an interpretation in terms of electroweak interactions
on leptons, suggesting that this approach may lead to models of fundamental particles with physically
relevant properties. In particular, the asymmetric nature of the octonionic multiplication table appears
to lead naturally9 to precisely three generations of particles, with single-helicity neutrinos, observed
properties of nature which as yet have no theoretical foundation.
In the present work, we use various Cartan decompositions of the Lie algebra e6 to further
extend this construction in two distinct ways. First, we identify additional real subalgebras of
sl(3, O), thus completing the explicit construction of nested chains of subgroups of S L(3, O) begun
in Ref. 8, building on the Lorentzian structure of S L(2, O). In particular, we locate the “missing”
C4 subgroups of S L(3, O) referred to in Ref. 8, in the form SU (3, 1, H). We then reinterpret the
Cartan decompositions used in our construction as (vector space) isomorphisms of the complexified
Lie algebra eC
6 , yielding an Abelian group of such isomorphisms that acts on the 5 real forms of
a) Electronic mail: awangberg@winona.edu
b) Electronic mail: tevian@math.oregonstate.edu
0022-2488/2013/54(8)/081703/14/$30.00
54, 081703-1
C 2013 AIP Publishing LLC
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A. Wangberg and T. Dray
J. Math. Phys. 54, 081703 (2013)
e6 , allowing us to identify 8 particular copies of these real forms, based on their relationship to the
original copy of sl(3, O).
We briefly review the basic structure of S L(3, O) in Sec. II A, and the basic properties of Cartan
decompositions in Sec. II B. We construct three important Cartan decompositions of e6 in Sec. III,
and use them to construct nested chains of subalgebras of sl(3, O) in Sec. IV, which are used in
Sec. IV B to construct a tower of subalgebras in sl(3, O) based upon our preferred basis of the Cartan
subalgebra, completing the program begun in Ref. 8. Finally, we explore the relationships between
the real forms of e6 in Sec. V.
II. THE BASICS
A. SL(3, O)
We summarize here the description of the Lie group S L(3, O) = E 6(−26) , as given in
Refs. 5,7,8, and 10. The Albert algebra H3 (O) consists of the 3 × 3 octonionic Hermitian matrices.
A complex matrix M is one whose components lie in some complex subalgebra of O; such matrices
act on X ∈ H3 (O) as
X −→ MX M†
(1)
and S L(3, O) can be defined as the (composition of) such transformations that preserve the determinant of X . Each Jordan matrix X can be decomposed as
X =
X θ
θ† ·
(2)
and M ∈ S L(2, O) ⊂ S L(3, O) acts on vectors X and spinors θ via the embedding
M −→ M =
M0
.
0 1
(3)
An explicit identification of elements of S L(2, O) with the (double cover of the) Lorentz group
SO(9, 1) was given in Ref. 6, naturally generalizing the identification of S L(2, C) with (the double
cover of) SO(3, 1). We refer to elements of S L(3, O) of the form (3) as being of type 1. Cyclically
permuting rows and columns in (3) results in analogous embeddings of type 2 (1 in the upper left) and
type 3 (1 in the middle). There are thus 3 natural embeddings of S L(2, O) sitting inside S L(3, O),
whose elements we label according to their type and character as a Lorentz transformation. Thus,
1
1
the 45 elements of type 1 S L(2, O) consist of 9 boosts Btq
and 36 independent rotations Rqr
, where
the spatial labels q, r run over x, z, and the imaginary octonionic units {i, j, k, k, j, i, }. Some
of these 45 generators of type 1 S L(2, O) require the use of more than one transformation of the
form (1), a phenomenon we call nesting.
The generators of the 3 natural copies of S L(2, O) do indeed span S L(3, O), but are not
independent, so we introduce a preferred set of generators. First of all, although there are 3 natural
chains of subgroups of the form G2 ⊂ SO(7) ⊂ SO(9, 1), the 3 copies of G2 are in fact the same,
so we dispense with the superscript labeling type, and call the 14 generators {Aq , Gq }, with q now
ranging over the imaginary octonionic units. We denote the remaining 7 generators of (type 1) SO(7)
1
, for a total
by Sq1 . Second, by triality there is only one copy of SO(8), so we add the 7 generators Rxq
of 28 generators so far. The remaining 24 rotations in S L(3, O) are the 3 types of rotations with z,
to which must be added the 27 − 1 = 26 independent boosts—the diagonal zt boosts are not all
independent.
Our 78 preferred generators of S L(3, O) become a basis of the Lie algebra sl(3, O) under
differentiation, denoted by a dot; this preferred basis is summarized in Table I. For further details,
see Refs. 5, 7, and 8.
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A. Wangberg and T. Dray
J. Math. Phys. 54, 081703 (2013)
TABLE I. Our preferred basis for sl(3, O).
Boosts
(2)
Ḃt1z
(3)
Ḃt1x
(21)
1
Ḃtq
Ḃt2z
Ḃt2x
2
Ḃtq
Ḃt3x
3
Ḃtq
(3)
Ṙx1z
(21)
1
Ṙzq
Ṙx2z
2
Ṙzq
Ṙx3z
3
Ṙzq
(7)
Ġ q
(7)
Ṡq1
(7)
1
Ṙxq
Simple
rotations
Transverse
rotations
(7)
Ȧq
B. Graded Lie algebras
A Z2 -grading of a Lie algebra g is a decomposition
g=p⊕m
(4)
such that
[p, p] ⊂ p
[m, m] ⊂ p
(5)
[p, m] ⊂ m
so that p is itself a Lie algebra, but m is merely a vector space. We will assume further that m is not
Abelian (in which case it would be an ideal of g), and that the Killing form is nondegenerate on each
of p and m.11 Each Z2 -grading defines a map θ on g, given by
θ (P + M) = P − M
(P ∈ p, M ∈ m),
(6)
which is clearly an involution, that is, a Lie algebra automorphism whose square is the identity map.
Conversely, an involution θ defines a Z2 -grading in terms of the eigenspaces of θ with eigenvalues
± 1, which can be shown to satisfy (5).12
A Cartan decomposition of a real Lie algebra g is a Z2 -grading of g, with the further property
that the Killing form B is negative definite on p and positive definite on m. In this case, (p,m) is
called a Cartan pair, the signature of g is (|p|,|m|), and the associated involution θ is called a Cartan
involution. Informally, p consists of rotations, and m of boosts.
We extend this terminology to the complexification gC of g. Whereas each real form of gC
admits a unique (up to isomorphism) Cartan decomposition, gC itself will admit several, one for
each real form.
A slight modification of an involution θ on g can be used to map one real form of gC to another.
Use θ to split g into eigenspaces p and m as above, and then introduce the associated Cartan map
φ* on gC via
φ ∗ (P + M) = P + ξ M
(P ∈ pC , M ∈ mC )
(7)
with ξ 2 = − 1, that is, where ξ is a square root of − 1 which commutes with all imaginary units
used in the representation of g. The structure constants of a real form g of gC are real by definition,
and since
[p, p] ⊂ p
[p, ξ m] ⊂ ξ m
[ξ m, ξ m, ] ⊂ ξ 2 p = (−1)p
(8)
then p ⊕ ξ m also has real structure constants, and is therefore also a real form of the complex Lie
algebra gC . Note that φ ∗ is a vector space isomorphism, but not a Lie algebra isomorphism—as must
be the case if it takes one real form to another. Further information regarding the interplay between
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081703-4
A. Wangberg and T. Dray
J. Math. Phys. 54, 081703 (2013)
involutive automorphisms, the Killing form, and real forms of a complex Lie algebra may be found
in Ref. 12.
We claim that any Z2 -grading of a real Lie algebra g is in fact the Cartan map associated with
(the restriction of) some Cartan decomposition of gC .
Lemma: Every Z2 -grading of a real Lie algebra g is the restriction of a Cartan decomposition
of gC . Equivalently, the extension of the Z2 -grading to gC is the image of a Cartan decomposition
of some (other) real form of gC under an associated Cartan map.
Proof: Let g = p ⊕ m be a Z2 -grading of g. Extend this grading to the Z2 -grading
gC = pC ⊕ mC of gC , then restrict each component to the compact real form gc of gC . Since
the compactness of g depends entirely on the signature of the Killing form, gc is clearly isomorphic
to g as a vector space, and we therefore obtain the vector space decomposition
gc = pc ⊕ mc .
(9)
Since we started with a Z2 -grading, and since gc is a Lie algebra by assumption, we must have
[pc , pc ] ⊂ pC ∩ gc = pc
(10)
with similar expressions holding for the other commutators in (5). Thus, (9) is a Z2 -grading of gc .
It is now straightforward to invert the associated Cartan map, constructing the vector space
g = pc ⊕ ξ mc
(11)
with ξ 2 = − 1. Even though associated Cartan maps are not Lie algebra isomorphisms, they do
preserve the Z2 -grading, so g is a Lie algebra, and the Killing form is negative definite on pc , and
positive definite on mc , by construction (and the assumed nondegeneracy of the Killing form). Thus,
(11) is a Cartan decomposition of the real form g of gC .
The Z2 -gradings of g therefore correspond to the possible real forms of gC . We will use
associated Cartan maps φ ∗ : gC → gC not only to identify different real forms of gC , but also to
identify real subalgebras of our particular real form g. When applied to the compact real form
gc = pc ⊕ mc , the map φ* changes the signature from (|p| + |m|, 0) to (|p|, |m|). Similar counting
arguments give the signature of φ ∗ (g) when g is non-compact, since φ* changes some compact
generators into non-compact generators, and vice versa. We then use the rank, dimension, and
signature of φ ∗ (g) to identify the particular real form of the algebra. Using tables of real forms of gC
showing their maximal compact subalgebras,12 we can also identify p = g ∩ φ ∗ (g) as a subalgebra
of our original real form g. Since the maximal compact subalgebra of φ ∗ (g) is known, we can further
identify its pre-image as a non-compact subalgebra of g.
III. CARTAN DECOMPOSITIONS OF sl(3, O)
A. Some gradings of sl(3, O)
We first make some comments about our preferred basis for sl(3, O), which is listed in
Table I and further discussed in Ref. 8. Let b and r be the vector subspaces consisting of boosts
and rotations, respectively. Our preferred choice of basis favors type 1 transformations, in the sense
that we choose to represent so(8), as well as its subalgebra g2 , in terms of transformations of type
1. Let t1 be the subspace spanned by Ḃt2z − Ḃt3z and all type 1 transformations, and t2 and t3 be the
subspaces spanned by the type 2 and type 3 transformations in our preferred basis which are not in
t1 .13 Let t23 = t2 ⊕ t3 . Finally, let h be the subspace corresponding to transformations that preserve
the preferred quaternionic subalgebra H = 1, k, k, , and let h⊥ be its orthogonal complement.
The vector space h is spanned by transformations with no labels in {i, j, j, i} and is in fact a
subalgebra, while h⊥ is the subspace spanned by transformations having one label from {i, j, j, i};
1
, and Ḃt3x , while h⊥ contains orthogonalh contains quaternionic basis elements, such as Ȧk , Ṙz
3
1
quaternionic basis elements such as Ȧi , Ṙzi , and Ḃt j .
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A. Wangberg and T. Dray
J. Math. Phys. 54, 081703 (2013)
Direct computation shows that
[r, r] ⊂ r
[t1 , t1 ] ⊂ t1
[r, b] ⊂ b
[t1 , t23 ] ⊂ t23
[b, b] ⊂ r
[t23 , t23 ] ⊂ t1
[h, h] ⊂ h
h, h⊥ ⊂ h⊥
⊥ ⊥
h ,h ⊂ h
(12)
so that each of the decompositions r ⊕ b, t1 ⊕ t23 , and h ⊕ h⊥ is a Z2 -grading of sl(3, O). We
therefore introduce the involutions φ s , φ t , and φH on sl(3, O), given by
φs (R + B) = R − B
R ∈ r, B ∈ b
φt (T1 + T23 ) = T1 − T23
T1 ∈ t1 , T23 ∈ t23 ,
φH (H + H ) = H − H H ∈ h, H ∈ h⊥
(13)
∗
and let φs∗ , φt∗ , and φH
be the associated Cartan maps on eC
6 .
The associated Cartan map φs∗ transforms sl(3, O), which has signature (52, 26), into the
compact real form φs∗ (sl(3, O)), which has signature (78, 0). The subalgebra
φs∗ (sl(3, O)) ∩ sl(3, O) = r
(14)
has dimension 52 and is therefore easily seen to be the compact real form su(3, O) of f4 . For this
case, the compact part of φs∗ (sl(3, O)) is the entire algebra, so that its pre-image is already a known
subalgebra of sl(3, O), namely, sl(3, O) itself.
B. Some subalgebras of sl(3, O)
We obtain two interesting subalgebras of sl(3, O) when we apply the associated Cartan map
φt∗ . First, the signature of g = φt∗ (sl(3, O)) is again (52, 26), since t23 contains the same number
of boosts and rotations (16 of each). The real form g is therefore isomorphic to, but distinct from,
sl(3, O), with maximal compact subalgebra gc = f4 . Hence, the pre-image of gc is a real form of f4
in sl(3, O). It has signature (36, 16), and the 16 non-compact generators identify this real form of f4
as su(2, 1, O) = f4(−20) . The second subalgebra comes from looking at
φt∗ (sl(3, O)) ∩ sl(3, O) = t1 .
(15)
Because |t1 | = 46, it must be a real form of d5 ⊕ d1 .14 But t1 contains so(9, 1) = sl(2, O), which
has signature (36, 9), as well as the boost Ḃt2z − Ḃt3z . Writing u(−1) for the non-compact real
representation of d1 generated by Ḃt2z − Ḃt3z , we identify t1 as the subalgebra sl(2, O) ⊕ u(−1), with
signature (36, 9) ⊕ (0, 1).
∗
to sl(3, O). In this case, the signature of
We also obtain two subalgebras by applying φH
∗
⊥
g = φH (sl(3, O)) is (36, 42), since h contains 28 rotations and 12 boosts, resulting in a net of 16
changes in signature. The maximal compact subalgebra gc of g is su(4, H), the compact real form
of c4 . The pre-image of gc has signature (24, 12), and, due to the 12 non-compact generators, can be
identified as su(3, 1, H). The invariant subalgebra
∗
(sl(3, O)) ∩ sl(3, O) = h
φH
(16)
has dimension |h| = 38 and signature (24, 14), and is therefore a real form of a5 ⊕ a1 with signature
∗
, there are 21 which are quaternionic and form
(21, 14) ⊕ (3, 0). Of the 24 rotations unaffected by φH
the subalgebra su(3, H). The remaining three rotations Ak , Ak , A are the elements of g2 that leave
invariant the quaternionic subalgebra spanned by {k, , k}. The four elements i, j, i, and j can
be paired into two complex pairs (of which i + i, j + j is one choice), and the transformations
Ȧk , Ȧk , Ȧ act as su(2, C) transformations producing the other pairs of complex numbers. We
henceforth refer to this copy of su(2, C) as su(2)H . Hence, the 24 compact elements form the
algebra su(3, H) ⊕ su(2)H , and h = sl(3, H) ⊕ su(2)H .
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A. Wangberg and T. Dray
J. Math. Phys. 54, 081703 (2013)
TABLE II. Intersections of subspaces t1 , t23 , h, and h⊥ .
∩
t1
t23
∩
h
h⊥
h1
h⊥
1
h23
h⊥
23
h
h⊥
Subspace
t1
t23
(16, 6)
(8, 8)
(20, 4)
(8, 8)
Signature
C. More subalgebras of sl(3, O)
∗
⊥
Consider now the composition φt∗ ◦ φH
. We define the subspaces h1 , h23 , h⊥
1 , and h23 of sl(3, O)
⊥
to be the intersections of pairs of subspaces t1 , t23 , h, and h , as indicated in Table II. For example,
⊥
h1 = h ∩ t1 and h⊥
23 = h ∩ t23 . Table II also indicates the number of basis elements which are
boosts and rotations in each of these spaces, whose commutation rules are given in Table III.
We see from Table III that, in addition to the subalgebras h = h1 ⊕ h23 and t1 = h1 ⊕ h⊥
1
∗
∗
constructed previously, h1 and h1 ⊕ h⊥
23 are also subalgebras of sl(3, O). Furthermore, φt ◦ φH fixes
∗
(sl(3, O)) ∩ sl(3, O) = h1 ⊕ h⊥
φt∗ ◦ φH
23
(17)
as a subalgebra, since everything in h1 is fixed by both maps, while everything in h⊥
23 is multiplied by
ξ 2 = − 1. Thus, even though the composition of associated Cartan maps is not quite an associated
Cartan map itself (due to the minus sign), it does lead to another Z2 -grading; we return to this point
in Sec. V A below.
The subalgebra p in (17) has dimension 38 and signature (24, 14), and is thus a real form of
a5 ⊕ a1 . However, it has a fundamentally different basis from h = sl(3, H) ⊕ su(2)H , as p uses a
mixture of the quaternionic transformations of type 1 with the orthogonal-quaternionic transformations of type 2 and type 3, while sl(3, H) is comprised of only the quaternionic transformations. We
therefore refer to this algebra as sl(2, 1, H) ⊕ su(2)2 . Explicitly,
1
su(2)2 = Ġ k + 2 Ṡk1 , Ġ k + 2 Ṡk
, Ġ + 2 Ṡ1 (18)
again corresponds to permutations of {i, j, j, i} and fixes {k, k, }, but is not in g2 . We also note
∗
(sl(3, O)) has dimension 36, and its pre-image
that the maximal compact subalgebra gc of φt∗ ◦ φH
in sl(3, O) has signature (24, 12). We identify this subalgebra as su(3, 1, H)2 , since this real algebra
has a different basis than our previously identified su(3, 1, H), which we henceforth refer to as
su(3, 1, H)1 .
We summarize in Table IV the subalgebras constructed from our three associated Cartan maps,
as well as compositions of these maps. For each map φ*, the second column lists the signature
of g = φ ∗ (sl(3, O)), the third column identifies, and gives the signature of, the fixed subalgebra
p = φ ∗ (sl(3, O)) ∩ sl(3, O), and the fourth column identifies the pre-image of the maximal compact
subalgebra of g = φ ∗ (sl(3, O)), listing both the algebra and its signature. Each algebra listed in the
third and fourth columns of Table IV is a subalgebra of sl(3, O), constructed using the other real
forms of e6 .
⊥
Despite recognizing h1 , h1 ⊕ h⊥
23 , h1 ⊕ h23 , and h1 ⊕ h1 as subalgebras of sl(3, O) using the
commutation relations in Table III, we note that su(3, 1, H)2 is not any of these subalgebras. In
Sec. IV, we use another technique involving associated Cartan maps to give a finer refinement of
⊥
TABLE III. Commutation structure of h1 , h23 , h⊥
1 , and h23 .
[,]
h1
h1
h1
h23
h23
h23
h⊥
1
h⊥
23
h23
h⊥
1
h⊥
23
h1
h⊥
23
h⊥
1
h⊥
1
h⊥
1
h⊥
23
h1
h23
h⊥
23
h⊥
23
h⊥
1
h23
h1
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081703-7
A. Wangberg and T. Dray
J. Math. Phys. 54, 081703 (2013)
TABLE IV. Compositions of associated Cartan maps and the corresponding
subalgebras of sl(3, O).
Signature of
Signature of
g = φ ∗ sl(3, O) p = g ∩ sl(3, O)
Map
1
(52, 26)
φs∗
(78, 0)
φt∗
(52, 26)
∗
φH
(36, 42)
φt∗ ◦ φs∗
(46, 32)
∗ ◦ φ∗
φH
s
(38, 40)
∗
φt∗ ◦ φH
(36, 42)
∗ ◦ φ∗
φt∗ ◦ φH
s
(38, 40)
Signature of
(φ ∗ )−1 (gc )
(52, 26)
(52, 0)
sl(3, O)
su(3, O)
(52, 0)
(52, 26)
su(3, O)
sl(3, O)
(36, 10)
(36, 16)
sl(2, O) ⊕ u(−1)
su(2, 1, O)
(24, 14)
(24, 12)
su(3, 1, H)1
sl(3, H) ⊕ su(2)H
(36, 16)
(36, 10)
su(2, 1, O)
sl(2, O) ⊕ u(−1)
(24, 12)
(24, 14)
su(3, 1, H)1
sl(3, H) ⊕ su(2)H
(24, 14)
(24, 12)
su(3, 1, H)2
sl(2, 1, H) ⊕ su(2)2
(24, 12)
(24, 14)
sl(2, 1, H) ⊕ su(2)2
su(3, 1, H)2
subspaces of sl(3, O), allowing us to provide a nice basis for su(3, 1, H)2 and other subalgebras of
sl(3, O).
IV. CONSTRUCTING SUBALGEBRAS OF sl(3, O)
A. Using composition of associated Cartan maps
We have already used associated Cartan maps to identify the maximal subalgebras of sl(3, O).
This was done by separating the algebra into two separate spaces, one of which was left invariant by
the map. In this section, we use composition of associated Cartan maps to separate sl(3, O) into four
or more subspaces spanned by either the compact or non-compact generators, with the condition
that the map either preserves the entire subspace or changes the character of all the basis elements
in the subspace. We identify additional subalgebras of sl(3, O) by taking various combinations of
these subspaces.
∗
, as well as the subspaces r, b,
We continue to use the associated Cartan maps φs∗ , φt∗ , and φH
⊥
t1 , t23 , h, and h defined in Sec. III.
∗
◦ φs∗ . This map fixes the subspaces rH = r ∩ h, consisting
We first consider the composition φH
of quaternionic rotations, as well as b⊥ = b ∩ h⊥ , consisting of orthogonal-quaternionic boosts.
∗
◦ φs∗ , the two subspaces bH and r⊥ = r ∩ h⊥ change signature. These spaces consist of
Under φH
orthogonal-quaternionic rotations and quaternionic boosts, respectively. The dimensions of these
four spaces are displayed in Table V.
∗
◦ φs∗ , and can thus identify subalgebras of
We list the signature of these spaces under φH
∗
∗
φH ◦ φs (sl(3, O)). However, we are primarily interested in the pre-image of these subalgebras in our
∗ ◦ φ∗.
TABLE V. Splitting of e6 basis under φH
s
∩
h
h⊥
r
b
|rH | = 24
|bH | = 14
|r⊥ | = 28
|b⊥ | = 12
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081703-8
A. Wangberg and T. Dray
J. Math. Phys. 54, 081703 (2013)
TABLE VI. Splitting of e6 basis under φt∗ ◦ φs∗ .
∩
t1
t23
r
b
|r1 | = 36
|b1 | = 10
|r23 | = 16
|b23 | = 16
preferred algebra sl(3, O). Determining the signature of these spaces in sl(3, O) is straightforward,
as the rotations are compact and the boosts are not.
We use the subspaces represented in Table V to identify subalgebras of sl(3, O). As previously
identified, the 24 rotations in rH fixed by the automorphism form the subalgebra su(3, H)1 ⊕ su(2)H ,
where su(3, H)1 ⊂ su(3, 1, H)1 . The entries in the first column of Table V represent all quaternionic
rotations and boosts, and form the subalgebra sl(3, H) ⊕ su(2)H . Of course, the entries in the first
row, rH and r⊥ , form f4 . We finally consider the entries rH and b⊥ on the main diagonal of the
table. Two orthogonal-quaternionic boosts commute to a quaternionic rotation, and an orthogonalquaternionic boost commuted with a quaternionic rotation is again an orthogonal-quaternionic boost.
Hence, the subspace rH ⊕ b⊥ closes under commutation and is a subalgebra with signature (24, 12).
While both so(9 − n, n, R) and su(4 − n, n, H) have dimension 36, only su(3, 1, H) has 12 boosts,
so rH ⊕ b⊥ is the previously identified su(3, 1, H)1 .
The algebra rH ⊕ b⊥ = su(3, 1, H)1 is a real form of the complex Lie algebra c4 . Since c4
contains c3 but not b3 , the 21-dimensional subalgebra contained within rH is a real form of c3 , not of
b3 . In addition, any simple 21-dimensional subalgebra of rH ⊕ b⊥ is a real from of c3 . Eliminating
the boosts from su(3, 1, H)1 leaves su(3, H)1 .
We next consider the composition φt∗ ◦ φs∗ . This automorphism separates the basis for sl(3, O)
into the subspaces
r1 = r ∩ t1
b1 = b ∩ t 1 ,
b23 = b ∩ t23
r23 = r ∩ t23.
(19)
As shown in Table VI, this map leaves the signatures of r1 and b23 alone, while it reverses the
signatures of r23 and b1 .
We again use the subspaces represented in Table VI to identify subalgebras of sl(3, O). The
subspace r1 is the subalgebra so(9, R), containing all subalgebras so(n, R) for n ≤ 9, and we
have already seen that the subspace t1 = r1 ⊕ b1 is so(9, 1, R) ⊕ u(−1). Again, the complete set
of rotations r1 ⊕ r23 form the subalgebra su(3, O), which is a real form of f4 . Interestingly, the
subspace r1 ⊕ b23 on the main diagonal is another form of f4 . The 16 boosts in r1 ⊕ b23 identify this
form of f4 as su(2, 1, O).
∗
◦ φs∗ , which creates a finer refinement than compoWe finally consider the composition φt∗ ◦ φH
sitions of two maps. The resulting subspaces are listed in Table VII. We continue with our previous
conventions for designating intersections of subspaces, that is, r23,⊥ = r ∩ t23 ∩ h⊥ .
Using this division of sl(3, O), we find a large list of subalgebras of sl(3, O) simply by
combining certain subspaces. The subspace description of these algebras, as well as their identity and
signature in sl(3, O), is listed in Table VIII. This fine refinement of sl(3, O) provides a description
of the basis for su(3, 1, H)1 and su(3, 1, H)2 , as well as sl(3, H) and sl(2, 1, H).
∗ ◦ φ∗.
TABLE VII. Splitting of e6 basis under φt∗ ◦ φH
s
∩
h1
h23
h⊥
1
h⊥
23
r
b
|r1,H | = 16
|b1,H | = 6
|r23,H | = 8
|b23,H | = 8
|r1,⊥ | = 20
|b1,⊥ | = 4
|r23,⊥ | = 8
|b23,⊥ | = 8
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081703-9
A. Wangberg and T. Dray
J. Math. Phys. 54, 081703 (2013)
TABLE VIII. Subalgebras of sl(3, O) using Cartan decompositions.
Subalgebra of sl(3, O)
Signature
su(2, H) ⊕ su(2)H ⊕ su(2)
su(3, H)2 ⊕ su(2)H
sl(2, H) ⊕ su(2)H
⊕su(2) ⊕ u(−1)
su(2, 1, H)1 ⊕ su(2)H
su(2, 1, H)2 ⊕ su(2)H
su(3, H) ⊕ su(2)H
so(5, R) ⊕ so(4, 1, R)
so(9) = su(2, O)
sl(3, H) ⊕ su(2)H
(10 + 3 + 3, 0)
(21 + 3, 0)
(10 + 3 + 3, 5 + 1)
Basis
r1,H
r1,H ⊕ r23,⊥
h1 = r1,H ⊕ b1,H
r1,H ⊕ b23,⊥
r1,H ⊕ b23,H
rH = r1,H ⊕ r23,H
r1,H ⊕ b1,⊥
r1 = r1,H ⊕ r1,⊥
h = r1,H ⊕ b1,H
⊕b23,H ⊕ r23,H
h1 ⊕ h⊥
23 = r1,H ⊕ b1,H
⊕r23,⊥ ⊕ b23,⊥
t1 = r1,H ⊕ b1,H
⊕b1,⊥ ⊕ r1,⊥
rH ⊕ b⊥ = r1,H ⊕ b23,⊥
⊕ r23,H ⊕ b1,⊥
r1,H ⊕ r23,⊥
⊕b23,H ⊕ b1,⊥
r = r1,H ⊕ r23,⊥
⊕r23,H ⊕ r1,⊥
r1 ⊕ b23 = r1,H ⊕ b23,⊥
⊕ b23,H ⊕ r1,⊥
(13 + 3, 8)
(13 + 3, 8)
(21 + 3, 0)
(10 + 6, 4)
(36, 0)
(21 + 3, 14)
sl(2, 1, H)1 ⊕ su(2)2
(21 + 3, 14)
sl(2, O) ⊕ u(−1)
(36, 9 + 1)
su(3, 1, H)1
(24, 12)
su(3, 1, H)2
(24, 12)
su(3, O)
(52, 0)
su(2, 1, O)
(36, 16)
B. Chains of subalgebras of sl(3, O)
We have used associated Cartan maps to produce large simple subalgebras of sl(3, O), ranging
in dimension from 52, for f4 , to 21, for c3 . Each of these subalgebras in turn has its own associated
Cartan maps, which we could use to find even smaller subalgebras, thereby giving a catalog of
subalgebra chains contained within sl(3, O). However, having identified the real form of the large
subalgebras of sl(3, O), it is not too difficult a task to find the smaller algebras simply by looking
for simple subalgebras of smaller dimension and/or rank, using the tables of real forms listed
in Ref. 12 when needed. Furthermore, we can choose our smaller subalgebras and their bases
so that they use a subset of our preferred basis for the Cartan subalgebra of sl(3, O), namely
1
, Ṡ1 , Ġ , Ȧ }, henceforth referred to as our Cartan basis.
{ Ḃt1z , Ḃt2z − Ḃt3z , Ṙx
We display these chains of subalgebras of sl(3, O) in the following tables. Each table is built
from a u(1) algebra, generated by a single element of our Cartan basis. We extend each algebra g to
a larger algebra g by adding elements to the basis for g. In particular, each algebra of higher rank
must add new elements of our Cartan basis, as indicated along the arrows (with dots suppressed).
Figure 1 is built from the algebra u(1) = G l − Sl1 , leading to
u(1) ⊂ su(1, H) ⊂ su(2, H) ⊂ su(3, H)1 ⊂ sl(3, H).
(20)
Additional subalgebras of sl(3, O) can be inserted into this chain of subalgebras. For instance,
we can insert sl(2, H) between su(2, H) and sl(3, H), and extend sl(2, H) to sl(2, O). We can also
expand su(1, H) to su(1, O) = so(7) and insert the so(n, R) chain for n ≥ 7 into the figure. However,
so(7) uses a basis in this chain that is not compatible with g2 = aut(O). We do not list all of the
1
into the chain and extend it to
possible u(1) subalgebras, but do add the subalgebra u(1) = Rx
su(2, C)s and sl(2, C)s , which use the standard (type 1) matrix definition of su(2, C) and sl(2, C).
With one exception, the algebras in Figure 1 are built from subalgebras by extending the basis
at each step; we do not allow any changes of basis. The one exception is the inclusion of the chain
su(2, C)s ⊂ su(3, C)s ⊂ su(3, H)1
(21)
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081703-10
A. Wangberg and T. Dray
J. Math. Phys. 54, 081703 (2013)
FIG. 1. Preferred subalgebra chains of e6 .
using the Cartan basis elements Ṡ1 and Ġ . We also use different notations to indicate possible
methods used to identify the subalgebras, with dashed and solid arrows indicating that the root
diagram of the smaller algebra can be obtained as a slice or as a projection of that of the larger
algebra; for details, see Ref. 15.
Finally, we have identified four different real forms of c3 , all of which contain su(2, H).
Space constraints limit us to listing only su(2, 1, H)1 and su(3, H)1 in Figure 1, but the algebras
su(2, 1, H)2 , su(3, H)2 , and su(3, 1, H)2 should also be in this table. We list these four real forms
of c3 algebras, all built from su(2, H), in Figure 2, and include all the algebras which are built from
the c3 algebras. Figure 2 can be incorporated into Figure 1 without having to adjust our choice of
Cartan basis.
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081703-11
A. Wangberg and T. Dray
J. Math. Phys. 54, 081703 (2013)
FIG. 2. Four real forms of c3 . An underline indicates the presence of su(2)2 , an overline indicates the presence of su(2)H ,
each initial vertical line indicates the presence of one component of b23,H ⊕ r23,⊥ , and each final vertical line indicates the
presence of one component of r23,H ⊕ b23,⊥ .
V. DISCUSSION
A. The group of associated Cartan maps
We return to the structure of associated Cartan maps (7). The square of φ* is clearly the original
involution φ, and the inverse of φ* is obtained by replacing ξ with − ξ , or equivalently as the cube
of φ*. But the composition of two associated Cartan maps is not quite an associated Cartan map as
we have defined them above, although it does lead to a graded Lie algebra structure, which we use
to define a group operation as follows.
Given two associated Cartan maps, written symbolically as
φ1∗ (p1 + m1 ) = p1 + ξ m1 ,
(22)
φ2∗ (p2 + m2 ) = p2 + ξ m2 ,
(23)
p1 ⊕ m1 = gC = p2 ⊕ m2
(24)
(φ1∗ φ2∗ )(q pp + q pm + qmp + qmm ) = q pp + ξ q pm + ξ qmp + qmm ,
(25)
where
we define their product to be
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081703-12
A. Wangberg and T. Dray
J. Math. Phys. 54, 081703 (2013)
FIG. 3. Composition of associated Cartan maps of e6 acting on real forms of e6 , showing the maximal compact subalgebra
under φ s .
where
q pp ∈ p1 ∩ p2
q pm ∈ p1 ∩ m2
qmp ∈ m1 ∩ p2
qmm ∈ m1 ∩ m2
(26)
and which differs from composition by the sign of the last term. It is easily verified that
gC = (q pp ⊕ qmm ) ⊕ (q pm ⊕ qmp )
(27)
is a Z2 -grading of gC , so that φ1∗ φ2∗ is indeed an associated Cartan map. The operation is
commutative, and φ φ is the identity map for any φ.
The set of associated Cartan maps therefore forms a group under the operation . We consider
∗
, which is easily
in particular the group generated by the associated Cartan maps φs∗ , φt∗ , and φH
3
seen to be a copy of (Z2 ) , and hence of order 8. The orbit of sl(3, O) = e6(52,26) under this group is
shown in Figure 3, from which the multiplication table can be inferred.
The multiplication table of the finite group (Z2 )3 is identical to that of the octonionic units,
without the minus signs, and can therefore also be represented using the 7-point projective plane.
Using commutation as the operation, the same multiplication table applies directly to the 8 subspaces
listed in Table VII, as shown in Figure 4. Using octonionic language for this multiplication table,
the 15 proper subalgebras of sl(3, O) listed in Table VIII consist precisely of 1 “real” subalgebra,
corresponding to the “identity element” r1,H , 7 “complex” subalgebras, formed by adding any
one other subspace, corresponding to the “points” in the multiplication table, and 7 “quaternionic”
subalgebras, formed by adding any one additional subspace (and ensuring the algebra closes),
corresponding to the “lines” in the multiplication table. There is of course one subalgebra missing
from this description, namely the “octonionic” algebra sl(3, O) itself, corresponding to the entire
Fano “plane.”
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081703-13
A. Wangberg and T. Dray
J. Math. Phys. 54, 081703 (2013)
FIG. 4. The (Z2 )3 multiplication table under commutation for the subspaces of e6 determined by the associated Cartan maps
∗ , and φ ∗ . All elements square to r
φt∗ , φH
1,H .
s
B. Real forms of e6
There are 5 real forms of e6 , all of which appear in Figure 3, although it is at first sight somewhat
surprising that several of them appear more than once. However, our interpretation of S L(3, O) is tied
to a particular choice of basis, so different copies of a given real form yield different decompositions
of sl(3, O), not necessarily with the same signature.
We regard Figure 3 itself as an indication that there are really 8 “real forms” of e6 of relevance
to the structure of S L(3, O), and hence of possible relevance to physics. At the very least, all 8 of
these real forms played a role in the construction of the “maps” of sl(3, O) given in Figures 1 and 2.
ACKNOWLEDGMENTS
This paper is a revised version of Chap. 5 of a dissertation submitted by A.W. in partial fulfillment
of the degree requirements for his Ph.D. in Mathematics at Oregon State University.15 The revision
was made possible in part through the support of a grant from the John Templeton Foundation.
1 P.
Jordan, “Über die Multiplikation quantenmechanischer Größen,” Z. Phys. 80, 285–291 (1933).
Jordan, J. von Neumann, and E. Wigner, “On an algebraic generalization of the quantum mechanical formalism,” Ann.
Math. 35, 29–64 (1934).
3 A. A. Albert, “On a certain algebra of quantum mechanics,” Ann. Math. 35, 65–73 (1934).
4 H. Georgi, Lie Algebras in Particle Physics (Benjamin Cummings, Reading, MA, 1982).
5 T. Dray and C. A. Manogue, “Octonions and the Structure of E ,” Comment. Math. Univ. Carolin. 51, 193–207 (2010),
6
http://cmuc.karlin.mff.cuni.cz/pdf/cmuc1002/dray.pdf; e-print arXiv:0911.2255.
6 C. A. Manogue and J. Schray, “Finite Lorentz transformations, automorphisms, and division algebras,” J. Math. Phys. 34,
3746–3767 (1993); e-print arXiv:hep-th/9302044.
7 C. A. Manogue and T. Dray, “Octonions, E , and particle physics,” J. Phys.: Conf. Ser. 254, 012005 (2010); e-print
6
arXiv:0911.2253.
8 A. Wangberg and T. Dray, “E , the Group: The structure of S L(3, O);” e-print arXiv:1212.3182.
6
2 P.
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081703-14
A. Wangberg and T. Dray
J. Math. Phys. 54, 081703 (2013)
9 T. Dray, and C. A. Manogue, “Quaternionic spin,” in Clifford Algebras and Mathematical Physics, edited by R. Abłamowicz
and B. Fauser (Birkhäuser, Boston, 2000), pp. 29–46; e-print arXiv:hep-th/9910010.
of this material also appears in Ref. 15.
11 For an example of a Z -grading that does not satisfy these conditions, see Ref. 8.
2
12 R. Gilmore, Lie Groups, Lie Algebras, and Some of Their Applications (Wiley, 1974), reprinted by Dover Publications,
Mineola, New York, 2005.
13 Since Ḃ 1 + Ḃ 2 + Ḃ 3 = 0, and since t contains Ḃ 1 and Ḃ 2 − Ḃ 3 , we regard both Ḃ 2 and Ḃ 3 as being elements of t .
1
1
tz
tz
tz
tz
tz
tz
tz
tz
14 The labels a , b , c , and d refer to the standard, Cartan-Killing classification of (complex!) simple Lie algebras, as, of
m
m m
m
course, do e6 , f4 , and g2 .
15 A. Wangberg, “The structure of E ,” Ph.D. thesis (Oregon State University, 2007), available at http://xxx.lanl.gov/
6
abs/0711.3447.
10 Much
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